Important Issues on Spectral Properties of a Transmission Eigenvalue Problem
Nowadays, inverse scattering is an important field of interest for many mathematicians who deal with partial differential equations theory, and the research in inverse scattering is in continuous progress. There are many problems related to scattering by an inhomogeneous media. Here, we study the tr...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2021/5795940 |
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| _version_ | 1832556472732483584 |
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| author | Besiana Cobani Aurora Simoni Ledia Subashi |
| author_facet | Besiana Cobani Aurora Simoni Ledia Subashi |
| author_sort | Besiana Cobani |
| collection | DOAJ |
| description | Nowadays, inverse scattering is an important field of interest for many mathematicians who deal with partial differential equations theory, and the research in inverse scattering is in continuous progress. There are many problems related to scattering by an inhomogeneous media. Here, we study the transmission eigenvalue problem corresponding to a new scattering problem, where boundary conditions differ from any other interior problem studied previously. more specifically, instead of prescribing the difference Cauchy data on the boundary which is the classical form of the problem, we consider the case when the difference of the trace of the fields is proportional to the normal derivative of the field. Typical concerns related to TEP (transmission eigenvalue problem) are Fredholm property and solvability, the discreteness of the transmission eigenvalues, and their existence. In this article, we provide answers for all these concerns in a given interior transmission problem for an inhomogeneous media. We use the variational method and a very important theorem on the existence of transmission eigenvalues to arrive at the conclusion of the existence of the transmission eigenvalues. |
| format | Article |
| id | doaj-art-6e1ac929c9ae4d5493e93561215deab6 |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-6e1ac929c9ae4d5493e93561215deab62025-02-03T05:45:22ZengWileyInternational Journal of Differential Equations1687-96431687-96512021-01-01202110.1155/2021/57959405795940Important Issues on Spectral Properties of a Transmission Eigenvalue ProblemBesiana Cobani0Aurora Simoni1Ledia Subashi2Department of Mathematics, Faculty of Natural Science, University of Tirana, Tirana, AlbaniaDepartment of Applied Mathematics, Faculty of Natural Science, University of Tirana, Tirana, AlbaniaDepartment of Mathematics, Faculty of Natural Science, University of Tirana, Tirana, AlbaniaNowadays, inverse scattering is an important field of interest for many mathematicians who deal with partial differential equations theory, and the research in inverse scattering is in continuous progress. There are many problems related to scattering by an inhomogeneous media. Here, we study the transmission eigenvalue problem corresponding to a new scattering problem, where boundary conditions differ from any other interior problem studied previously. more specifically, instead of prescribing the difference Cauchy data on the boundary which is the classical form of the problem, we consider the case when the difference of the trace of the fields is proportional to the normal derivative of the field. Typical concerns related to TEP (transmission eigenvalue problem) are Fredholm property and solvability, the discreteness of the transmission eigenvalues, and their existence. In this article, we provide answers for all these concerns in a given interior transmission problem for an inhomogeneous media. We use the variational method and a very important theorem on the existence of transmission eigenvalues to arrive at the conclusion of the existence of the transmission eigenvalues.http://dx.doi.org/10.1155/2021/5795940 |
| spellingShingle | Besiana Cobani Aurora Simoni Ledia Subashi Important Issues on Spectral Properties of a Transmission Eigenvalue Problem International Journal of Differential Equations |
| title | Important Issues on Spectral Properties of a Transmission Eigenvalue Problem |
| title_full | Important Issues on Spectral Properties of a Transmission Eigenvalue Problem |
| title_fullStr | Important Issues on Spectral Properties of a Transmission Eigenvalue Problem |
| title_full_unstemmed | Important Issues on Spectral Properties of a Transmission Eigenvalue Problem |
| title_short | Important Issues on Spectral Properties of a Transmission Eigenvalue Problem |
| title_sort | important issues on spectral properties of a transmission eigenvalue problem |
| url | http://dx.doi.org/10.1155/2021/5795940 |
| work_keys_str_mv | AT besianacobani importantissuesonspectralpropertiesofatransmissioneigenvalueproblem AT aurorasimoni importantissuesonspectralpropertiesofatransmissioneigenvalueproblem AT lediasubashi importantissuesonspectralpropertiesofatransmissioneigenvalueproblem |